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Evaluating Mutual Fund Performance within the Stochastic Discount Framework

Jonathan Fletcher

in Mutual Funds and Exchange-Traded Funds: Building Blocks to Wealth

Some view performance measures based on the stochastic discount factor approach as having a stronger theoretical basis than traditional performance measures. The stochastic discount factor approach ... More

Some view performance measures based on the stochastic discount factor approach as having a stronger theoretical basis than traditional performance measures. The stochastic discount factor approach can be used to evaluate mutual fund performance using linear factor models, nonlinear models, and measures based on weaker economic restrictions than required by a full asset pricing model. The stochastic discount factor approach can address investor heterogeneity, as investors can evaluate the value added by a fund differently from one another. This chapter provides an overview of the stochastic discount factor approach to evaluate mutual fund performance. The chapter discusses alternative approaches used to construct stochastic discount factors to evaluate fund performance and reviews empirical evidence on mutual fund performance using the stochastic discount factor approach.Less

MODERN RECURSIVE EQUILIBRIUM AND THE BASIC PRICING EQUATION

Patrick L. Anderson

in The Economics of Business Valuation: Towards a Value Functional Approach

The author introduces the “recursive” model that has emerged within micro-economics over the past few decades. This modern recursive equilibrium model is contrasted with the neoclassical model, in ... More

The author introduces the “recursive” model that has emerged within micro-economics over the past few decades. This modern recursive equilibrium model is contrasted with the neoclassical model, in terms of the optimization and time periods involved. The modern, multi-period consumer savings problem is introduced, as well as the “cake eating” problem and basic pricing equation. The author argues these form the basis of a modern microeconomic theory, and that the stochastic discount factor that emerges from the basic pricing equation provides a valuable insight that is lacking in the neoclassical and classical worlds. As with other valuation principles, the author tests the principle as a practical valuation tool for three actual businesses, demonstrating that is provides an incomplete basis for valuation of private firms.Less

Asset Pricing Theories, Models, and Tests

Nikolay Gospodinov and Cesare Robotti

in Portfolio Theory and Management

An important but still partially unanswered question in the investment field is why different assets earn substantially different returns on average. Financial economists have typically addressed ... More

An important but still partially unanswered question in the investment field is why different assets earn substantially different returns on average. Financial economists have typically addressed this question in the context of theoretically or empirically motivated asset pricing models. Since many of the proposed “risk” theories are plausible, a common practice in the literature is to take the models to the data and perform “horse races” among competing asset pricing specifications. A “good” asset pricing model should produce small pricing (expected return) errors on a set of test assets and should deliver reasonable estimates of the underlying market and economic risk premia. This chapter provides an up-to-date review of the statistical methods that are typically used to estimate, evaluate, and compare competing asset pricing models. The analysis also highlights several pitfalls in the current econometric practice and offers suggestions for improving empirical tests. Examples of topics include stochastic discount factor, two-pass cross-sectional regressions, misspecified models, and useless factors.Less

Factor Theory

Andrew Ang

in Asset Management: A Systematic Approach to Factor Investing

Assets earn risk premiums because they are exposed to underlying factor risks. The capital asset pricing model (CAPM), the first theory of factor risk, states that assets that crash when the market ... More

Assets earn risk premiums because they are exposed to underlying factor risks. The capital asset pricing model (CAPM), the first theory of factor risk, states that assets that crash when the market loses money are risky and therefore must reward their holders with high risk premiums. While the CAPM defines bad times as times of low market returns, multifactor models capture multiple definitions of bad times across many factors and states of nature.Less

Potentials and Positive Interest

Tomas Björk

in Arbitrage Theory in Continuous Time

In this chapter we describe two approaches to interest rate theory which are built on probabilistic potential theory. This approach leads to positive interest rates and there is a nice connection to ... More

In this chapter we describe two approaches to interest rate theory which are built on probabilistic potential theory. This approach leads to positive interest rates and there is a nice connection to the stochastic discount factor. We present two alternatives: the Flesaker–Hughston approach, and the Rogers approach.Less

The Martingale Approach to Arbitrage Theory

Tomas Björk

in Arbitrage Theory in Continuous Time

In this chapter the theoretical level is substantially increased, and we discuss in detail the deep connection between financial pricing theory and martingale theory. The first main result of the ... More

In this chapter the theoretical level is substantially increased, and we discuss in detail the deep connection between financial pricing theory and martingale theory. The first main result of the chapter is the First Fundamental Theorem which says that the market is free of arbitrage if and only if there exists an equivalent martingale measure. We provide a guided tour through the Delbaen–Schachemayer proof and we then apply the theory to derive a general risk neutral pricing formula for an arbitrary financial derivative. We also discuss the Second Fundamental Theorem which says that the market is complete if and only if the martingale measure is unique. We define the stochastic discount factor and use it to provide an alternative form of the pricing formula. Finally, we provide a summary for the reader who wishes to go lighter on the (rather advanced) theory.Less

Continuous-Time Markets

Kerry E. Back

in Asset Pricing and Portfolio Choice Theory

A continuous‐time model of a securities market is introduced. The intertemporal budget constraint is defined. SDF processes and prices of risks are defined and characterized. Many properties of SDF ... More

A continuous‐time model of a securities market is introduced. The intertemporal budget constraint is defined. SDF processes and prices of risks are defined and characterized. Many properties of SDF process are analogous to those in a single‐period model, including the relation to the risk‐free rate, orthogonal projections, the Hansen‐Jagannathan bound, and factor pricing. To value future cash flows using an SDF process, we need to assume a local martingale is a martingale. Sufficient conditions including Novikov’s condition are discussed. Use of the martingale representation theorem in a complete market to derive a portfolio that replicates a payoff is explained. A Markovian model is introduced, in which the investment opportunity set depends on state variables that form a Markov process.Less

Stochastic Discount Factors

Kerry E. Back

in Asset Pricing and Portfolio Choice Theory

SDFs are defined. The first order condition for portfolio choice is interpreted as: an investor’s marginal rate of substitution is an SDF. There is a strictly positive SDF if and only if there are no ... More

SDFs are defined. The first order condition for portfolio choice is interpreted as: an investor’s marginal rate of substitution is an SDF. There is a strictly positive SDF if and only if there are no arbitrage opportunities, and there is some SDF if and only if the law of one price holds. There is a unique SDF if and only if the market is complete. Orthogonal projections are defined. There is a unique SDF in the span of the assets, and it equals the projection of any SDF onto the span of the assets. The Hansen‐Jagannathan bounds are derived. Orthogonal projections are used to show how an investor with quadratic utility hedges labor income risk.Less

Asset Pricing III: Arbitrage-Based Pricing

Harold L. Cole

in Finance and Financial Intermediation: A Modern Treatment of Money, Credit, and Banking

This chapter develops a arbitrage-based pricing model in which the extent to which the stochastic discount factor varies is unrestricted.

The Cox–Ingersoll–Ross Factor Model

Tomas Björk

in Arbitrage Theory in Continuous Time

In this chapter we present a slightly simplified version of the general Cox–Ingersoll–Ross factor model. We derive the equilibrium short rate, the equilibrium stochastic discount factor, the relevant ... More

In this chapter we present a slightly simplified version of the general Cox–Ingersoll–Ross factor model. We derive the equilibrium short rate, the equilibrium stochastic discount factor, the relevant market prices of risk, and the equilibrium Girsanov kernel. A central planner is also introduced, and we prove equivalence.Less

Endowment Equilibrium: Unit Net Supply

Tomas Björk

in Arbitrage Theory in Continuous Time

We now move on to study dynamic equilibrium within the framework of a unit net supply endowment model. This model is studied in detail, beginning with the simplest scalar model and ending with a ... More

We now move on to study dynamic equilibrium within the framework of a unit net supply endowment model. This model is studied in detail, beginning with the simplest scalar model and ending with a multifactor non-Markovian vector model. We use the martingale approach from Chapter 27 to analyze the model and we derive explicit formulas for the equilibrium stochastic discount factor and the equilibrium short rate.Less

Dynamic Securities Markets

Kerry E. Back

in Asset Pricing and Portfolio Choice Theory

The dynamic model with time‐additive utility is defined. The intertemporal budget constraint is explained. SDF processes are defined in terms of a martingale property. There is a strictly positive ... More

The dynamic model with time‐additive utility is defined. The intertemporal budget constraint is explained. SDF processes are defined in terms of a martingale property. There is a strictly positive SDF process if and only if there are no arbitrage opportunities. Dynamic complete markets are explained. The difference between the price of an asset and its value calculated from an SDF process is called a bubble. There is no bubble if a transversality condition is satisfied. Some constraints on trading strategies are needed to rule out Ponzi schemes. SDF processes are derived for nominal asset prices and for asset prices denominated in a foreign currency.Less

Utility Indifference Pricing and Other Topics

Tomas Björk

in Arbitrage Theory in Continuous Time

In this chapter we discuss two methods of pricing in incomplete markets, based on utility functions. This theory comes in the shape of a global and a local version. Both versions are discussed, and ... More

In this chapter we discuss two methods of pricing in incomplete markets, based on utility functions. This theory comes in the shape of a global and a local version. Both versions are discussed, and for the local version we connect to the theory of stochastic discount factors and equilibrium theory.Less

Factor Models

Kerry E. Back

in Asset Pricing and Portfolio Choice Theory

The CAPM and factor models in general are explained. Factors can be replaced by the returns or excess returns that are maximally correlated (the projections of the factors). A factor model is ... More

The CAPM and factor models in general are explained. Factors can be replaced by the returns or excess returns that are maximally correlated (the projections of the factors). A factor model is equivalent to an affine representation of an SDF and to spanning a return on the mean‐variance frontier. The use of alphas for performance evaluation is explained. Statistical factor models are defined as models in which factors explain the covariance matrix of returns. A proof is given of the Arbitrage Pricing Theory, which states that statistical factors are approximate pricing factors. The CAPM and the Fama‐French‐Carhart model are evaluated relative to portfolios based on sorts on size, book‐to‐market, and momentum.Less

Equilibrium Theory: A Simple Production Model

Tomas Björk

in Arbitrage Theory in Continuous Time

This is the first of several chapters dealing with the dynamic equilibrium theory. As an instructive first example we study a simple Cox–Ingersoll–Ross type of production model. The equilibrium ... More

This is the first of several chapters dealing with the dynamic equilibrium theory. As an instructive first example we study a simple Cox–Ingersoll–Ross type of production model. The equilibrium concept is given a precise formulation and we derive the equilibrium short rate as well as the equilibrium stochastic discount factor. We also study the associated optimization problem for a central planner and prove that this is equivalent to the equilibrium problem.Less